Determinant conjecture for globally positive group matrices

Let $G$ be a group and let $A\in\operatorname{Mat}_n(\mathbb{Z}[G])$ be a globally positive matrix, meaning a matrix to which the stated positivity condition applies. Determinant Conjecture. The Fuglede--Kadison determinant satisfies

$$\det_{\mathcal{N}(G)}(r^{(2)}(A))\geq 1.$$

The source states that this conjecture implies the Approximation Conjecture through a bounded Fuglede--Kadison determinant lemma, and attributes the relevance of this implication to Schick. Its resolution is not specified here.